A new mathematical preprint reports that, among convex regions on a sphere with a fixed area, the round disk has the greatest cartographic distortion when mapped onto a flat plane. The result addresses whether the round shape is the worst-case geometry for every convex region with the same area.
This is a theorem about geometric objects, not a test of maps or map users. It considers a closed convex set Ω on the unit sphere and a round disk Dρ with equal area. Under those conditions, the paper uses the disk as the distortion benchmark and reaches a deterministic mathematical conclusion.
A benchmark for stretching and compression
For any equal-area closed convex region, the paper states that a map into the plane exists with a lower Lipschitz constant of 1 and an upper constant given by the displayed sinρ ρ term. In ordinary language, these constants set bounds on how much distances can change when the curved region is represented on a flat surface.
The same sinρ ρ expression is reported as the optimal benchmark for maps from the round disk itself. The preprint’s central comparison is therefore that no equal-area convex region needs a worse distortion bound than the disk benchmark, making the disk the stated worst case within this class.
The author presents the construction as a solution to Milnor’s cartography problem. The introduction also reports that equality for the disk determines the corresponding map up to rescaling. But strictly lower distortion for every non-round region is not presented as a proved result in the supplied text; it appears instead as Exercise 7.1.
How the proof builds the map
The proof begins by constructing a finite subdivision of the region into ρ-balanced sets whose coastlines can be made arbitrarily short. The pieces are connected through a shared tree-like structure, giving the argument a way to combine local maps into one global construction.
Within each suitable slice, the argument uses a foliation, or a family of chords arranged to cover the slice. Under the stated balance and chord-length conditions, this produces a fishbone map with the sinρ ρ Lipschitz bound. The same conditions give a backbone map with Lipschitz constant 1.
The fishbone and backbone component maps agree along the shared tree, so they assemble into global maps. The paper states that the resulting backbone is 1-Lipschitz and acts as a left inverse of the planar map, written as b ◦ f = idΩ. Since ε can be chosen arbitrarily, the proof then applies the Arzelà–Ascoli theorem to obtain a limiting map with the stated Lipschitz constants.
The boundaries of the result
The theorem is restricted to closed convex subsets of the unit sphere that satisfy the equal-area condition. In the proof, convexity gives ρ ≤ π/2, while the main argument assumes ρ < π/2. The remaining hemisphere case is handled with polar coordinates.
The work is a preprint, identified in its front matter as arXiv:2608.20265v1 [math.DG] and dated 20 Aug 2026. It uses deterministic geometric propositions, chord constructions, Jacobi-field calculations and Lipschitz estimates rather than an empirical sample or statistical analysis.
The result does not establish that every non-round region has strictly lower distortion. Nor does the supplied text give an explicit coordinate formula for the limiting map: the map is obtained through an Arzelà–Ascoli existence argument. The analysis also leaves open the listed structural properties of optimal maps and possible extensions to nonconvex domains and related constant-curvature manifolds.
The supplied text includes acknowledgments to named collaborators but does not report a funding source. Conflicts of interest, data availability and code availability are not reported in the supplied analysis.
Paper data and sources
Original title: Milnor's cartography problem
Authors: Anton Petrunin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text